Why am I getting the incorrect value of lambda?
$TD(\lambda)$ return has the following form:
\begin{equation}
G_t^\lambda = (1 - \lambda) \sum_{n=1}^{\infty} \lambda^{n-1} G_{t:t+n}
\end{equation}
For you MDP $TD(1)$ looks like this:
\begin{align}
G &amp;= 0.64 (r_0 + r_2 + r_4 + r_5 + r_6) + 0.36(r_1 + r_3 + r_4 + r_5 + r_6)\\
G &amp;\approx 6.164
\end{align}
$TD(\lambda)$ looks like this:
\begin{equation}
G_0^\lambda = (1-\lambda)[\lambda^0 G_{0:1} + \lambda^1 G_{0:2} + \lambda^2 G_{0:3} + \lambda^3 G_{0:4} + \lambda^4 G_{0:5} ]
\end{equation}
Now each $G$ term separately:
\begin{align}
G_{0:1} &amp;= 0.64(r_0 + v_1) + 0.36(r_1 + v_2) \approx 7.864\\
G_{0:2} &amp;= 0.64(r_0 + r_2 + v_3) + 0.36(r_1 + r_3 + v_3) \approx -5.336\\
G_{0:3} &amp;= 0.64(r_0 + r_2 + r_4 + v_4) + 0.36(r_1 + r_3 + r_4 + v_4) \approx 25.864\\
G_{0:4} &amp;= 0.64(r_0 + r_2 + r_4 + r_5 + v_5) + 0.36(r_1 + r_3 + r_4 + r_5 + v_5) \approx -11.936\\
G_{0:5} &amp;= 0.64(r_0 + r_2 + r_4 + r_5 + r_6 + v_6) + 0.36(r_1 + r_3 + r_4 + r_5 + r_6 
 + v_6) \approx -0.336
\end{align}
Finally, we need to find $\lambda$ so that the return is equal to $TD(1)$ return, we have:
\begin{equation}
6.164 = (1 - \lambda)[7.864 - 5.336\lambda + 25.864\lambda^2 - 11.936\lambda^3 - 0.336\lambda^4]
\end{equation}
When you solve this equation, one of the solutions is  $0.205029$ which is close to what you needed to get considering the numerical errors. Your problem was that you only considered probability in first state, but that decision prolongs to future states as well.

EDIT 

As pointed out by bewestphal, this is not a full solution, it misses one crucial step to get it fully correct. Hint for it can be found in his answer and that's the correct solution.